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Available online at www.sciencedirect.com

Fluid Phase Equilibria 260 (2007) 190–194

Modeling vapor–liquid equilibria of ethanol+ 1,1,1,2,3,3,3-heptafluoropropane binary

mixtures using PC-SAFT

Matthias Kleiner, Gabriele Sadowski ∗Lehrstuhl fur Thermodynamik, Universitat Dortmund, Emil-Figge-Str. 70, D-44227 Dortmund, Germany

Received 31 January 2007; received in revised form 11 April 2007; accepted 15 April 2007Available online 20 April 2007

bstract

One objective of the Third Industrial Fluid-Properties Simulation Challenge was the extrapolation of mixture phase equilibrium informa-ion obtained from one isotherm to other temperatures. In this work, we present modeling results for the vapor–liquid equilibrium of thethanol/1,1,1,2,3,3,3-heptafluoropropane (HFC-227ea) mixture obtained with the PC-SAFT equation of state. The required pure-component

arameters were determined by fitting pure-component vapor pressures and liquid density data. To get a quantitative description of the mix-ure, a temperature-independent binary interaction parameter was adjusted to experimental bubble-point data at 283.17 K. Using these parameters,ubble-point pressures at T = 343.13 K were predicted for different concentrations.

2007 Elsevier B.V. All rights reserved.

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eywords: PC-SAFT; Equation of state; Cross-association; Phase equilibrium;

. Introduction

The ability to predict or to correlate thermodynamic prop-rties and phase equilibria with equations of state remains anmportant goal for the simulation of chemical processes and sub-equent process development. Especially, modeling of systemsontaining polar or associating compounds requires a physicallyased model which is able to account for the different types ofntermolecular interactions like association (hydrogen bonding)r dipolar interactions.

Significant progress towards the development of physicallyased equations of state was made by applying principles oftatistical mechanics. Many recent studies account for the non-pherical molecular shape by assuming molecules to be chainsf freely jointed spherical segments [1–7]. Many of these molec-lar models were further refined by specifically accounting for

ssociation or for dipolar interactions.

One example for these developments is the PC-SAFT equa-ion of state which has successfully been applied to a broad

∗ Corresponding author. Tel.: +49 231 755 2635; fax: +49 231 755 2572.E-mail address: G.Sadowski@bci.uni-dortmund.de (G. Sadowski).

2

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378-3812/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2007.04.015

ofluorocarbon; Alcohol

ange of substances and mixtures including simple fluids with-ut specific interactions [8–10], associating fluids [11] as wells dipolar and quadrupolar components and their mixtures12–17]. An excellent performance of the PC-SAFT model waslso shown for various polymer as well as copolymer systems18–29].

However, since cross-interactions between unlike moleculesre not considered, very asymmetric mixtures were still diffi-ult to treat with SAFT-based equations of state. This applies,or example, to mixtures where at least one of the componentsoes not self-associate but is able to form hydrogen bonds withnother component in the mixture (which we refer to as “inducedssociation”). To account for this type of cross-association inhe ethanol/HFC-227ea mixture, a simple approach is appliedere which does not require the adjustment of any additionalarameters.

. Theory

The PC-SAFT equation of state is based on a perturbationheory in which the hard chain serves as the reference fluid. Theerturbations account for contributions of dispersive attractionnd association interactions. The reduced residual Helmholtz

hase

e

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σ

ε

cD

wt

X

Δ

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Hmia

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•

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M. Kleiner, G. Sadowski / Fluid P

nergy is thus written as the sum of these contributions

Ares

NkT= Ahc

NkT+ Adisp

NkT+ Aassoc

NkT(1)

here k and T are the Boltzmann constant and the tempera-ure, respectively. N is the total number of molecules. In Eq.1), Ahc/NkT is the expression for the Helmholtz energy of theard-chain reference fluid and Adisp/NkT denotes the dispersionontribution of chain molecules. For a detailed description ofhese contributions, the reader is referred to earlier publications8]. Non-associating components can be described using threeure-component parameters: the segment number m, the seg-ent diameter σ and the energy parameter ε/k. These parameters

re determined by simultaneously fitting liquid densities andapor pressure data of the pure component. The mixture parame-ers are estimated by conventional Berthelot–Lorentz combiningules where one adjustable binary interaction parameter kij cane used to correct the mixture dispersion energy:

ij = 12 (σi + σj) (2)

ij = √εiεj(1 − kij) (3)

The strong short-range interactions of associating fluids areonsidered by an additional association contribution Aassoc/NkT.ue to Wertheim [30–33] and Chapman et al. [3,4] it is given as

Aassoc

NkT=

∑i

xi

∑Ai

(lnXAi − XAi

2+ 1

2

)(4)

here XAi is the fraction of molecules i that are not bonded athe association site A and can be calculated as

Ai =⎛⎝1 + ρ

∑j

xj

∑Bj

XBj ΔAiBj

⎞⎠

−1

(5)

The association strength ΔAiBi can be approximated as [4]

AiBj = ghsij (dij)κAiBj σ3

ij

(exp

(εAiBj

kT

)− 1

)(6)

here ghsij (dij) is the pair-distribution function of hard spheres.

he parameters εAiBj and κAiBj describe the associationtrength and the effective volume of an association site,espectively. These parameters are simultaneously adjusted toure-component data.

To describe mixtures of two associating substances, thetrength of the cross-associating interactions εAiBj and κAiBj

etween unlike molecules is estimated by applying combiningules as suggested by Wolbach and Sandler [34]

AiBj = 12 (εAiBi + εAjBj ) (7)

AiBj = √κAiBiκAjBj

( √σiiσjj

)3

(8)

(1/2)(σii + σjj

However, mixtures where induced-association may occur areifficult to treat with SAFT-based equations of state. Since thessociation parameters εAiBiand κAiBi are only available for

T

c(

Equilibria 260 (2007) 190–194 191

he self-associating component, no cross-association parame-ers can be determined by using the combining rules (Eqs. (7)nd (8)).

This also applies to the system ethanol/HFC-227ea whereFC-227ea does not self-associate but act as proton donor in theixture with ethanol. To account for these induced-association

nteractions in this mixture, we apply here the following simplepproach:

The association parameter εAiBiof HFC-227ea is set to zero.This ensures that no association is modeled for pure HFC-227ea.The association-volume parameter κAiBi of HFC-227ea isassumed to be equal to the value of the associating componentin the mixture (here ethanol).

Based on these settings, the cross-association parameters inhe mixture can now be obtained by applying the conventional

ixing rules (Eqs. (7) and (8)). As a result, the cross-associationnergy parameter calculated by Eq. (7) is thus half of thealue of the associating component ethanol. We follow here aethodology that was also used by Folas et al. [35] to account

or the solvation in mixtures of associating fluids with aro-atic or olefinic compounds. However, in contrast to their

pproach we do not treat the volume parameter for the cross-ssociation κAiBi as an adjustable parameter but calculate thisarameter directly from the respective pure-component param-ter of the associating component. This is a pragmatic approachut it does not need any additional adjustable parameters andherefore improves the predictive capability of the model. Theeneral applicability of this approach was confirmed in an exten-ive study where numerous binary and ternary mixtures werenvestigated [36]. In this study, it was demonstrated that theredictive capability as well as the ability of PC-SAFT to corre-ate mixture phase equilibria quantitatively could be improvedonsiderably. However, due to the nature of the simulationhallenge we focus here solely on the system ethanol/HFC-27ea.

. Results

.1. Model parameters

The pure-component parameters used within the frame-ork of PC-SAFT are determined by fitting pure-componentapor pressure and liquid density data. Since ethanol is anssociating component, five pure-component parameters areequired. These parameters were earlier determined by Grossnd Sadowski [11] assuming two association sites for ethanolccording to the 2B association scheme proposed by Huangnd Radosz [5]. These parameters as well as the averagebsolute deviations (AAD) for liquid densities and vaporressures in the considered temperature range are listed in

able 1.

Since HFC-227ea does not self-associate, only three pure-omponent parameters mi, σi and εi were fitted to liquid densitiesGruzdev et al. [37]) and to vapor pressure data (Shi et al. [38]).

192 M. Kleiner, G. Sadowski / Fluid Phase Equilibria 260 (2007) 190–194

Table 1Pure-component parameters for ethanol

Component i Mi (g/mol) mi σi (A) εi/k (K) εAiBi /k (K) κAiBi AAD% T range (K) Ref.

psat ρ

Ethanol 46.069 2.3827 3.1771 198.24 2653.4 0.032384 0.99 0.79 230–516 [11]

Table 2Pure-component parameters for HFC-227ea

Component i Mi (g/mol) mi σi (A) εi/k (K) εAiBi /k (K) κAiBi a AAD% T range (K) Data ref.

psat ρ

H –

OT u

Ttb

mtatTpvtias

3

r

FT(m

escfrppaiib0

tTi

FC-227ea 170.03 3.5190 3.3165 164.18

a The value of κAiBi for HFC-227ea was set to the value of ethanol and was N

he parameters are given in Table 2. The AAD values reveal thathe pure-component properties of HFC-227ea can be describedy the PC-SAFT model with good accuracy.

To account for the induced-association interactions in theixture, we apply the above-described approach and set

he association parameter εAiBi of HFC-227ea to zero. Thessociation-volume parameter κAiBi was assumed to be equal tohe value of the associating component ethanol (see Table 2).hese parameters are simply used in addition to the threeure-component parameters of HFC-227ea that were fitted pre-iously. The cross-association parameters in the mixture arehen calculated by applying the mixing rules for associat-ng components (Eqs. (7) and (8)). Similar to ethanol, it wasssumed that each HFC-227ea molecule has two associationites.

.2. Mixture phase equilibria

This section presents results for binary vapor–liquid equilib-ia of ethanol/HFC-227ea mixtures obtained with the PC-SAFT

ig. 1. Vapor–liquid equilibrium of the ethanol–HFC-227ea mixture at= 283.17 K. Dashed line, prediction via PC-SAFT from pure-component data

kij = 0); solid line, PC-SAFT correlations with kij = −0.012. Points are experi-ental bubble-point pressure data [39].

ts

d

FTwp

0.032384 0.26 0.44 243–375 [37,38]

sed for the determination of the pure-component parameters.

quation of state. In Fig. 1 experimental bubble-point pres-ures from Chai Kao and Schiller [39] at T = 283.17 K areompared with calculations using the approach to accountor induced-association as described above. The dashed lineepresents a prediction of the phase behavior obtained fromure-component data only without fitting a binary interactionarameter (kij = 0). It can be seen that the PC-SAFT model isble to predict the phase behavior qualitatively. The solid linen Fig. 1 further demonstrates that the model can be broughtnto excellent quantitative agreement with the experimental datay introducing a small binary interaction parameter (kij = −.012).

Using the same parameters, the bubble-point pressures ofhe ethanol/HFC-227ea mixture were predicted at T = 343.13 K.he calculated vapor-liquid phase behavior as well as the exper-

mental bubble-point data [39] (that were not available at the

ime of the calculations) are depicted in Fig. 2. The results areummarized in Table 3.

To quantify the predicted results of the PC-SAFT model, theeviation of the calculated bubble-points from the experimental

ig. 2. Vapor–liquid equilibrium of the ethanol–HFC-227ea mixture at= 343.13 K. Prediction with PC-SAFT. The binary parameter (kij = −0.012)as determined by fitting VLE data at 283.17 K. Points are experimental bubble-oint pressure data [39].

M. Kleiner, G. Sadowski / Fluid Phase

Table 3Bubble-point pressures of the ethanol–HFC-227ea mixture at T = 343.13 K

xi,ethanol pi,exp (bar) pi,calc (bar) �pi (%)

0.0000 14.870 14.854 0.110.0604 14.220 14.326 −0.740.1228 13.660 13.867 −1.510.3314 12.220 12.691 −3.860.5219 10.970 11.384 −3.770.7260 8.859 8.728 1.480.8547 6.165 5.7967 5.980.9440 3.215 2.934 8.731.0000 0.7274 0.7182 1.28

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X

x

Gε

ε

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ρ

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xperimental values pi,exp [39], predicted cloud-point pressures pi,calc usingC-SAFT (kij = −0.012 was determined using the data at 283.17 K) and relativeeviations �pi of experimental data points and predictions.

ata was calculated based on the following equation:

= 100

n

n∑i=1

∣∣∣∣pi,exp − pi,calc

pi,exp

∣∣∣∣ (9)

here pi,exp and pi,calc are the experimental and calculatedubble-point pressures of data point i, respectively. Using thealues of Table 3 (excluding the vapor pressures of the pure com-onents), the average deviation F of experimental and calculatedubble-point pressures at 343.13 K is 3.72%. This investigationeveals that the PC-SAFT model is well suited not only for theuantitative description of phase equilibria (as at 283 K) but islso a predictive tool for save extrapolations to other tempera-ures.

. Conclusion

The PC-SAFT equation of state was applied to model theapor–liquid equilibrium of the mixture of ethanol and HFC-27ea. To account for induced-association interaction in theixture, a simple approach is applied which does not require any

dditional adjustable parameter. Predictions of the phase behav-or using only pure-component parameters are in qualitativeood agreement with experimental data. Very accurate resultsan be obtained when a small binary interaction parameters introduced. Using these constant (temperature-independent)arameters, predictions at a higher temperature show an excel-ent agreement with experimental data with a mean averageeviation of 3.72%.

ist of symbolsHelmholtz energy (J)temperature-dependent segment diameter (A)deviation of experimental and calculated values (%)

hs radial distribution function of the hard-sphere fluid (−)Boltzmann constant (J/K)

ij binary interaction parameter (−)

number of segments per chain (−)molar mass (g/mol)total number of molecules (−)temperature (K)

[

[[

Equilibria 260 (2007) 190–194 193

fraction of monomer units not bonded to the associationsite (−)

i mole fraction of component i (−)

reek letters/k temperature-independent attraction energy (K)AiBi /k association energy (K)AiBi association volume (−)

total number density of molecules (A−3)temperature-independent segment diameter (A)

ubscript, j component

uperscriptsssoc contribution due to association,B association sitesisp dispersion contributionc hard-chain contribution

cknowledgements

The authors are grateful to the Deutsche Forschungsgemein-chaft for supporting this work with grant SAD 700/10.

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